On Ilmanen's multiplicity-one conjecture for mean curvature flow with type-I mean curvature

In this paper, we show that if the mean curvature of a closed smooth embedded mean curvature flow in R^3 is of type-I, then the rescaled flow at the first finite singular time converges smoothly to a self-shrinker flow with multiplicity one. This result confirms Ilmanen's multiplicity-one conjecture under the assumption that the mean curvature is of type-I. As a corollary, we show that the mean curvature at the first singular time of a closed smooth embedded mean curvature flow in R^3 is at least of type-I.


Introduction
In this paper, we study finite time singularities of closed smooth embedded mean curvature flow in R 3 . A one-parameter family of hypersurfaces x(p, t) : Σ n → R n+1 is called a mean curvature flow, if x satisfies the equation where H denotes the mean curvature of the hypersurface Σ t := x(t)(Σ) and n denotes the outward unit normal of Σ t . In the previous paper [33], we proved that the mean curvature of (1.1) must blow up at the first finite singular time for a closed smooth embedded mean curvature flow in R 3 . This paper can be viewed as a continuation of [33], and we will develop the techniques in [33] further to study the finite time singularities of mean curvature flow.

Singularities of mean curvature flow
The mean curvature flow with convexity conditions has been well studied during the past several decades. In [36], Huisken proved that if the initial hypersurface is uniformly convex, then after rescaling the mean curvature flow exists for all time and converges smoothly to a round sphere. When the initial hypersurface is mean-convex or two-convex, there are a number of estimates for the mean curvature flow (c.f. Huisken-Sinestrari [38][39], Haslhofer-Kleiner [34]), and these estimates are important to study the surgery of mean curvature flow(c.f. Huisken-Sinestrari [40], Brendle-Huisken [10], Haslhofer-Kleiner [35]). Moreover, for mean curvature flow with mean convex initial hypersurfaces, B. White gave some structural properties of the singularities in [58] [59], and B. Andrews also showed a noncollapsing estimate in [2]. However, all these results rely on convexity conditions of initial hypersurfaces, and it is very difficult to study general cases. For the curve shortening flow in the plane, following the work Gage [28] [29] and Gage-Hamilton [30] on convex curves Grayson [31] proved that any embedded closed curve in the plane evolves to a convex curve and subsequently shrinks to a point, and Andrews-Bryan [1] gave a direct proof of Grayson's theorem without using the monotonicity formula or classification of singularities. In the higher dimensions, we know very little results without convexity conditions. Colding-Minicozzi studied the generic singularities of the mean curvature flow in [19] [20]. For the classification of self-shrinkers without convexity conditions, S. Brendle [9] proved that the round sphere is the only compact embedded self-shrinkers in R 3 with genus 0, and L. Wang [55] showed that each end of a noncompact self-shrinker in R 3 of finite topology is smoothly asymptotic to either a regular cone or a self-shrinking round cylinder. However, it still remains wide open to understand the behavior of mean curvature flow at the singular time in the general cases.

The multiplicity-one conjecture and the main theorems
To study the singularities of mean curvature flow without convexity conditions, Ilmanen proposed a series of conjectures in [41] [42]. Suppose that the mean curvature flow (1.1) reaches a singularity at (x 0 , T ) with T < +∞. For any sequence {c j } with c j → +∞, we rescale the flow (1.1) by By Huisken's monotonicity formula [37] and Brakke's compactness theorem [3], a subsequence of Σ j t converges weakly to a limit flow T t , which is called a tangent flow at (x 0 , T ). In [41] Ilmanen showed that the tangent flow at the first singular time must be smooth for a smooth embedded mean curvature flow in R 3 , and he conjectured Conjecture 1.1. (Ilmanen [41] [42], the multiplicity-one conjecture) For a smooth one-parameter family of closed embedded surfaces in R 3 flowing by mean curvature, every tangent flow at the first singular time has multiplicity one.
Moreover, Ilmanen pointed out that the multiplicity-one conjecture implies a conjecture on the asymptotic structure of self-shrinkers in R 3 , and the latter conjecture has been confirmed recently by L. Wang [55]. If the initial hypersurface is mean convex or satisfies the Andrews condition, then the multiplicity-one conjecture holds (c.f. White [58], Haslhofer-Kleiner [34], Andrews [2]). Recently, A. Sun [52] proved that the generic singularity of mean curvature flow of closed embedded surfaces in R 3 modelled by closed self-shrinkers with multiplicity has multiplicity one. In general the multiplicityone conjecture is still wide open . In this paper, using the techniques from our previous work [33] we confirm the multiplicity-one conjecture under the assumption that the mean curvature is of type-I.
To state our result, we first introduce some notations. A hypersurface x : Σ n → R n+1 is called a self-shrinker, if x satisfies the equation for two constants D, Λ > 0. Then for any t i → +∞ there exists a subsequence of {Σ t i +t , −1 < t < 1} such that it converges in smooth topology to a complete smooth self-shrinker with multiplicity one as i → +∞.
In [33], we showed Theorem 1.3 under the assumption that the mean curvature decays exponentially to zero. In this special case, the flow (1.4) converges smoothly to a plane passing through the origin with multiplicity one. Theorem 1.3 means that under the assumption that the mean curvature is bounded for all time the flow (1.4) also converges smoothly to a self-shrinker with multiplicity one. In fact, Theorem 1.3 is not stated with the optimal condition. Checking the proof carefully, one can see that the conclusion of Theorem 1.3 still holds under the assumption that the mean curvature is uniformly bounded on any ball for all time: where C R is a constant depending on R. Note that if the flow (1.4) converges smoothly to a selfshrinker with multiplicity one, the condition (1.6) automatically holds by the self-shrinker equation. Thus, the condition (1.6) is also necessary for the smooth convergence of the flow (1.4). Therefore, we have reduced the solution of the multiplicity-one conjecture, i.e., Conjecture 1.1, to the examination of (1.5) or (1.6), which will be an interesting subject of study in the near future. The multiplicity-one conjecture is closely related to the extension problem of mean curvature flow. Huisken [36] proved that if the flow (1.1) develops a singularity at time T < ∞, then the second fundamental form will blow up at time T . A natural question is whether the mean curvature will blow up at the finite singular time of a mean curvature flow. Toward this question, A. Cooper [24] proved that |A||H| must blow up at the singular time of the flow. In a series of papers [44]- [46] Le-Sesum systematical studied this problem and they proved that the extension problem is true if the multiplicity-one conjecture holds, or the second fundamental form is of type-I at the singular time Furthermore, Le-Sesum [46] proved that the mean curvature is at least of type-I if (1.7) holds. Using Theorem 1.2, we can remove the type-I condition (1.7) of Le-Sesum's result as follows, which can also be viewed as an improvement of the extension theorem in [33].

Outline of the proof
Now we sketch the proof of Theorem 1.3. Assume that the mean curvature satisfies the type-I condition (1.3) along the flow (1.1) and the first singular time T < +∞. Then the mean curvature is uniform bounded along the rescaled flow (1.4). We have to show that the flow (1.4) converges smoothly to a self-shrinker with multiplicity one. The strategy is similar to [33], we first show a weak-compactness theorem and obtain the flow convergence is smooth away from a singular set. Then we use stability argument to remove the singular set. However, the technique here is much more involved. The proof consists of three steps: Step 1. Convergence of the rescaled mean curvature flow with multiplicities. In this step, since the mean curvature is uniformly bounded along the flow, we have the short-time pseudolocality theorem and the energy concentration property, and we can follow the arguments in [33] to develop the weak compactness theory of mean curvature flow. However, compared with [33], since the mean curvature doesn't decay to zero, we have the following difficulties: • No long time pseudolocality theorem; • The space-time singularities in the limit don't move along straight lines.
Because of lacking these results, we face a number of new technical difficulties to show the L-stability of the limit self-shrinker. These difficulties force us to use analysis tools to study the asymptotical behavior of the solution of the limit parabolic equation near the singular set.
Step 2. Show that the multiplicity of the convergence is one for one subsequence. As in [33], it suffices to show that the limit self-shrinker is L-stable. By the convergence of the flow away from the singular set, if every limit has multiplicity greater than one, we can renormalize the "heightdifference" function to obtain a positive solution of the equation ∂w ∂t = ∆w − 1 2 x, ∇w + |A| 2 w + 1 2 w, (1.8) away from the singular set. To show the L-stability of the limit self-shrinker, we have to show the following two estimates: • For each time, the asymptotical behavior of w is "good" near the singular set.
• Uniform L 1 norm of w independent of time.
By its construction, w is defined on any compact set away from the singular set and we have no estimates near the singular set by the geometric method. However, we found that w satisfies many good properties from the PDE point of view. In [7], Kan-Takahashi studied similar problem for some semilinear parabolic equations along time-dependent singularities in the Euclidean spaces. Kan-Takahashi showed their result for one time-dependent singularity, and the solution of the equation looks like log 1 r in dimension 2, where r is the distance from any point x to the singularity. However, in our case the solution of (1.8) may have multiple singularities, and these singularities may coincide at one point. Thus, we cannot apply Kan-Takahashi's result directly, and we need to develop their techniques to show that the solution w is in L 1 across the singularities and near the singular set the solution w roughly looks like where r k (x, t) denotes the intrinsic distance from a point x to a singularity ξ k (t) at time t. Here the constant c i may depend on t. In general, the L 1 norm of w may tend to infinity as t → +∞. In order to show uniform L 1 norm of w, we refine the argument in [33] and also use the estimate of w near the singularities to choose a sequence of time slices {t i }, and then we show that for such a special sequence the corresponding function w has uniform L 1 bound independent of t. Thus, for the special sequence t i , the auxiliary function w satisfies the two desired estimates. Then we can follow the argument in [33] to show that the convergence of (1.4) is smooth and of multiplicity one, for the special sequence {t i }.
Step 3. Show the multiplicity-one convergence for each subsequence. This step is a new difficulty beyond [33]. In [33], each limit, no matter what multiplicity it is, must be a flat plane passing through the origin. Therefore, up to rotation, different limits can be regarded as the same. By the monotonicity of the entropy, it is clear that if one limit is a multiplicity-one plane, then each limit must also be a multiplicity-one plane. However, in the current setting, each limit is only a self-shrinker and the limits may vary as the time sequences change. A priori, it is possible that one sequence converge to a multiplicity-one self-shrinker A, and the other sequence converge to a multiplicity two self-shrinker B = A. This possibility cannot be ruled out by only using the monotonicity of the entropy. To overcome this difficulty, we essentially use the smooth compactness theorem of self-shrinkers by Colding-Minicozzi [18]. Since the limit self-shrinkers form a compact set, we know that the local behavior of limit self-shrinkers are very close to that of planes on a fixed small scale. From this and the volume continuity, we derive an argument to show that the multiplicity is independent of the choice of subsequences. Therefore, every subsequence converges with multiplicity one.
It is interesting to know whether the above argument still works for the multiplicity-one conjecture without the mean curvature bound assumption (1.3). The main difficulty is the loss of pseudolocality result as in [33], since the points in the evolving surfaces may move drastically if the mean curvature is large. Furthermore, the loss of mean curvature bound also induces difficulties in applying PDE tools to analyze the singular set. However, as we discussed around (1.6), it is also logically possible to develop the estimate (1.6) directly.

Relation with other geometric flows
It is interesting to compare the mean curvature flow with the Ricci flow. The extension problem for Ricci flow has been extensively studied recently. Corollary 1.4 has a cousin theorem in the Ricci flow. In Theorem 1 of [54], it was shown that along the Ricci flow {(M, g(t)), 0 ≤ t < T } with the singular time T < +∞, we have which extends the famous Ricci extension theorem of N. Sesum [51]. Up to rescaling, the gap inequality (1.9) is equivalent to max M |Ric| g(t) ≥ δ along the rescaled Ricci flow solution It is easy to see that the scalar extension conjecture of the Ricci flow will hold automatically if one can prove the above inequality along the rescaled Ricci flow (1.10), just like the extension theorem of mean curvature in [33] follows directly from Corollary 1.4. The similarity between the regularity theory of rescaled mean curvature flow (1.4) and the rescaled Ricci flow (1.10) was noticed for a while. For example, such similarity was discussed in the introduction of [33]. Along the rescaled flows, the mean curvature bound condition (1.5) is comparable to the scalar curvature bound condition |R| ≤ C. Note that the Fano Kähler-Ricci flow provides many examples of the global solutions of the rescaled Ricci flow (1.10) and Perelman showed that |R| ≤ C holds automatically. The boundedness of the scalar curvature is crucial to study the convergence of Kähler-Ricci flow to a limit flow(c.f. Theorem 1.5 of [14], with journal version [15] and [16]). For time-slice convergence, see Tian-Zhang [53] for example. Since (1.5) is the comparable condition of Perelman's estimates, we can view Theorem 1.2 as the analogue of the convergence results in the Fano Käher-Ricci flow. However, we have to confess that we do not know any non-trivial examples satisfying the condition (1.5). By non-triviality we mean that we do not have positivity assumption of H. It will be very interesting to find out such examples.
The rescaled mean curvature flow can also be compared with the Calabi flow. In [4] E. Calabi studied the gradient flow of the L 2 -norm of the scalar curvature among Kähler metrics in a fixed cohomology class on a compact Kähler manifold, which is now well-known as the Calabi flow. X. X. Chen conjectured that the Calabi flow always exists globally for any initial smooth Kähler potential. Very recently, Chen-Cheng [5] proved that the Calabi flow exists as long as the scalar curvature is uniformly bounded. Therefore, to study the long time existence of Calabi flow, it is crucial to control the scalar curvature, which is similar to the mean curvature condition (1.5) for the rescaled mean curvature flow. Assuming the long time existence and the uniform boundedness of the scalar curvature, the current authors and K. Zheng showed the convergence of the Calabi flow in [43], just as Theorem 1.3 for rescaled mean curvature flow.

List of notations
In the following, we list the important notations in this paper.
• d(x, y) : the Euclidean distance from x to y. Defined in Definition 2.7.
• B r (p) : the open ball in R 3 centered at p with radius r. Defined in Definition 2.1.
• d g (x, y) : the intrinsic distance of (Σ, g) from x to y. First appears in the beginning of Section 4.
• B r (p) : the intrinsic geodesic ball in (Σ, g) centered at p with radius r. Defined in Definition 2.1.
• S : the space-time singular set. Defined in Proposition 2.8.
• S t = {x ∈ R 3 | (x, t) ∈ S}: the singular set at time t. Defined in Proposition 2.8.
• ξ(t) : a Lipschitz singular curve in S. First appears in Lemma 2.11.
• S I : the union of the singular set on a time interval I. Defined in (3.32).
• u i : the height difference function defined in (3.33).
• w i : the normalized difference function defined in (3.35).
• d H : the Hausdorff distance in the Euclidean space.
• r(x, t): the intrinsic distance function from x to the singular curve ξ(t). Defined in (4.22).
• r k (x, t): the intrinsic distance function from x to the singular curve ξ k (t). Defined in (3.125) and Theorem 4.2.
• M k,m (ρ, Ξ): a subset of a Riemannian manifold defined in Definition 4.1.
• Q r,t,t andQ r,t,t : the neighborhood of the singular curves. Defined in and (4.20) and (4.60).
• φ ξ : cutoff functions around the singular curves. Defined in Definition 4.7 and Definition 4.12.
• I ξ : a functional associated with a singular curve ξ. Defined in Definition 4.12.
• H(z) : a cutoff function defined in Definition 4.7. Note that the function H(z) is only used in Section 4. Since the mean curvature doesn't appear in Section 4, we keep the same notation H(z) as in [7].

The organization
The organization of this paper is as follows. In Section 2 we recall some facts on the pseudolocality theorem and energy concentration property. Moreover, we will show the weak compactness of mean curvature flow under some geometric conditions and we show the multiplicity of the convergence is a constant. In Section 3 we show the rescaled mean curvature flow with bounded mean curvature converges smoothly to a self-shrinker with multiplicity one, under the assumption that the auxiliary function satisfies good growth properties at the singular set. In Section 4 we will show the estimates of the auxiliary function by developing Kan-Takahashi's argument. Finally, we finish the proof of Theorem 1.2 in Section 5. In the appendices, we include two versions of the parabolic Harnack inequality and give the full details on the calculation of the linearized equation of rescaled mean curvature flow.

Acknowledgement
Bing Wang would like to thank Lu Wang for many helpful conversations.
2 Weak compactness of refined sequences 2.1 The pseudolocality theorem and energy concentration property In this subsection, we recall some results in [33]. First, we have the following definition.
Definition 2.1. (1). We denote by B r (p) the ball in R n+1 centered at p with radius r with respect to the standard Euclidean metric, and B r (p) ⊂ (M, g) the intrinsic geodesic ball on M centered at p with radius r with respect to the metric g.
Let {x 1 · · · , x n+1 } be the standard coordinates in R n+1 . Assume that x 0 = 0 and the tangent plane of Σ at x 0 is x n+1 = 0. Then there is a map with u(0) = 0 and |∇u|(0) = 0 such that the connected component containing Using Lemma 2.2, we show that the local area ratio of the surface is very close to 1.
For any δ > 0, there is a constant ρ 0 = ρ 0 (r 0 , δ) such that for any r ∈ (0, ρ 0 ) and any x ∈ B r 0 can be written as a graph of a function u over the tangent plane at x, which we assume to be P = Let r ∈ (0, ρ 0 ). We denote by Ω r the projection of C x (B r (x) ∩ Σ) to the plane P . Then for any x ′ ∈ ∂Ω r we have On the other hand, for any x ′ ∈ Ω ρ 0 we have the inequality Note that (2.2) and (2.3) imply that for any x ′ ∈ ∂Ω r , Thus, we haver := r 1 + 5184 where we used (2.3) and (2.6). Moreover, the volume ratio of C Combining (2.7) with (2.8), for any δ > 0 we can choose ρ 0 = ρ 0 (n, δ, r 0 ) further small such that (2.1) holds. The lemma is proved.
Next we recall the two-sided pseudolocality theorem in [33]. If the initial hypersurface can be locally written as a graph of a single-valued function, then we have the pseudolocality type results of the mean curvature flow by Ecker-Huisken [26] [27], M. T. Wang [56], Chen-Yin [23] and Brendle-Huisken [10]. However, in our case we have to apply the pseudolocality theorem for the hypersurfaces which may converge with multiplicities. Thus, we use the boundedness of the mean curvature to get the two-sided pseudolocality theorem in [33].

Weak compactness
As in [33], we use the pseudolocality theorem and the energy concentration property to develop the weak compactness of the mean curvature flow. Here we will replace the zero mean curvature condition in [33] by the boundedness of the mean curvature in the definition of refined sequences. The name of refined sequence originates from [13].
be a one-parameter family of closed smooth embedded surfaces satisfying the mean curvature flow equation (1.1). It is called a refined sequence if the following properties are satisfied for every i : (1) There exists a constant D > 0 such that d(Σ i,t , 0) ≤ D for any t ∈ (−1, 1), where d(Σ, 0) denotes the Euclidean distance from the point 0 ∈ R 3 to the surface Σ ⊂ R 3 and (2) There is a uniform constant Λ > 0 such that (3) There exists an increasing positive function ρ : R + → R + such that for any R > 0, (4) There is uniform N > 0 such that for all r > 0 and p ∈ R 3 we have (2.14) (5) There exist uniform constantsr, κ > 0 such that for any r ∈ (0,r] and any p ∈ Σ i,t we have Following the arguments as in minimal surfaces (c.f. White [57], or Colding-Minicozzi [17]), we have the weak compactness for mean curvature flow.
be a refined sequence. Then there exists a subsequence, still denoted by
As in [33], we show that the multiplicity in Proposition 2.8 is constant. To study the multiplicity, we define a function Then the multiplicity at (x, t) ∈ Σ ∞,t × (−1, 1) is defined by It is clear that m(x, t) is an integer. In the following result, we show that m(x, t) is independent of x and t. Note that in Lemma 3.14 of [33] we proved the same result under the assumption that the mean curvature decays exponentially to zero. The first two steps of the proof here are similar to that of [33] while the third step is different. We give all the details for completeness.
Lemma 2.9. Under the assumption of Proposition 2.8, the function m(x, t) is a constant integer on Σ ∞,t × (−1, 1). Namely, m(x, t) is independent of x and t.
Proof. The proof can be divided into three steps.
By Lemma 2.2, we can assume r 0 small such that B r 0 (x 0 ) ∩ Σ ∞,t 0 can be written as a graph over the tangent plane of Σ ∞,t 0 at x 0 . Let By Lemma 2.3, for any δ > 0 there exists ρ 0 = ρ 0 (r 0 , δ) ∈ (0, r 0 200 ) such that for any r ∈ (0, ρ 0 ) and any p ∈ B r 1 (x 0 ) ∩ Σ i,t 0 we have Suppose that B r 1 (x 0 ) ∩ Σ i,t 0 has m i connected components, where m i is an integer bounded by a constant independent of i by Proposition 2.8. After taking a subsequence of {Σ i,t 0 } if necessary, we can assume that m i are the same integer denoted by m with m ≥ 1. For any x ∈ B r 1 2 (x 0 ) ∩ Σ ∞,t 0 , we denote by α x the normal line passing through x of Σ ∞,t 0 . Since each component of B r 1 (x 0 ) ∩ Σ i,t 0 converges to B r 1 (x 0 ) ∩ Σ ∞,t 0 smoothly and B r 1 (x 0 ) ∩ Σ ∞,t 0 is a graph over the tangent plane of Σ ∞,t 0 at x 0 , α x intersects transversally each component of Σ i,t 0 at exactly one point. Suppose that Then (2.24) implies that for any integer j with 1 ≤ j ≤ m and any r ∈ (0, ρ 0 ), After shrinking r 0 if necessary, we can assume that B r (x) ∩ Σ ∞,t 0 has only one component for any r ∈ (0, r 1 2 ) and any x ∈ B r 1 2 (x 0 )∩Σ ∞,t 0 . Since for any 1 ≤ j ≤ m and r ∈ (0, ρ 0 ) we have p (j) In other words, for any x ∈ B r 1 2 (x 0 ) ∩ Σ ∞,t 0 and any r ∈ (0, ρ 0 ) we have By the connectedness of Σ ∞,t 0 \S t 0 , we know that m(x, t 0 ) is constant on Σ ∞,t 0 \S t 0 .
Step 2. For each t ∈ (−1, 1), m(x, t) is constant on Σ ∞,t . Fix t 0 ∈ (−1, 1). It suffices to consider a singular point p 0 ∈ S t 0 . Suppose that B r (p 0 ) ∩ Σ ∞,t 0 has no other singular points except p 0 for any r ∈ (0, r 0 ). Then all points in (B r (p 0 )\B ǫ (p 0 )) ∩ Σ ∞,t 0 are regular and (B r (p 0 )\B ǫ (p 0 )) ∩ Σ i,t 0 has m connected components. Thus, we have where we used (2.14) in the last inequality. Since each component of Note that m is also the multiplicity at each regular point in Σ ∞,t 0 by Step 1. Combining (2.28) with (2.29), we have (2.30) Taking ǫ → 0 in (2.30), we have This implies that the multiplicity of each singular point is the same as that of any regular point.

By
Step 2, (2.44) implies that for any x ∈ Σ ∞,t and y ∈ Σ ∞,t 0 we have Thus, the multiplicity m(x, t) is a constant independent of x and t. The lemma is proved.
To characterize the singular and regular points in Σ ∞,t , we have the following result.
Using the boundedness of the mean curvature and Lemma 2.10, we show that the singular set S consists of locally finitely many Lipschitz curves.
Proof. For any t 1 ∈ (−1, 1) and any p t 1 ∈ S t 1 ∩ B R (0), we show that there exists a Lipschitz curve in S passing through p t 1 . Since p t 1 is singular, by Lemma 2.10 we can find a sequence of points p i,t 1 ∈ Σ i,t 1 and r ′ = r ′ (Σ ∞,t 1 ) > 0 such that p i,t 1 → p t 1 and for any r ∈ (0, r ′ ), Let r 1 ∈ (0, r ′ ). By Lemma 3.4 in Li-Wang [33] and (2.62), there exists η(r 1 , Λ) > 0 such that for any t 2 ∈ (t 1 − η, t 1 + η) ∩ (−1, 1) we have where r 2 = r 1 + 2Λ|t 2 − t 1 |. Taking the limit in (2.63) and choosing η(r 1 , Λ) small, we have ). Since the mean curvature is uniformly bounded along the flow, all such that it converges to a point, which we denoted by p t 2 . Suppose that p t 2 is a regular point for some t 2 ∈ (t 1 − η, t 1 + η). Note that in Part (1) of Lemma 2.10, the constant r ′ depends only on the geometry of Σ ∞,t . Therefore, by Lemma 2.10 for δ = 1 4 we can choose a uniform constant Then (2.65) contradicts (2.64). Thus, p t 2 is a singular point. Moreover, we have By taking the limit i → +∞, we have Therefore, p t 1 lies in a Λ-Lipschitz curve in S. Since for any t ∈ (−1, 1) the set S t is locally finite by Proposition 2.8, the singular curves are locally finite. The lemma is proved.

The rescaled mean curvature flow
In this section, we will show the smooth convergence of rescaled mean curvature flow under uniform mean curvature bound. As is pointed out in the introduction, we have no long-time pseudolocality of the flow and the singularities don't move along straight lines. In order to show the L-stability of the limit self-shrinker, we need an estimate on the asymptotical behavior of the positive solution near the singular set (c.f. Lemma 3.21 and Lemma 3.28), and the proof of this estimate will be delayed to Section 4.
for two constants D, Λ > 0. Then for any t i → +∞ there exists a subsequence of {Σ t i +t , −1 < t < 1} such that it converges in smooth topology to a complete smooth self-shrinker with multiplicity one as i → +∞.
We sketch the proof of Theorem 3.1. First, we show the weak compactness for any sequence of the rescaled mean curvature flow in Lemma 3.4. Suppose that the multiplicity is at least two. By using the decomposition of spaces(c.f. Definition 3.5) we can select a special sequence {t i } in Lemma 3.13 for each ǫ > 0. This special sequence is needed to control the upper bound of the function w i away from the singular set by using the parabolic Harnack inequality (c.f. Lemma 3.16). Then we can take the limit for the function w i and obtain a positive function w with uniform bounds on any compact set away from the singular set(c.f. Lemma 3.17). The function w satisfies the linearized mean curvature flow equation. To study the growth behavior of w near the singular set, we take a sequence of ǫ i → 0 and for each ǫ i we repeat the above process to get a sequence of functions {w i,k } ∞ k=1 . After choosing a diagonal sequence and taking the limit, we get a function w with good growth estimates near the singular set (c.f. Proposition 3.23) by assuming Theorem 4.2 in the next section. The bounds of w imply the L-stability of the limit self-shrinker (c.f. Lemma 3.25), and this step also relies on Theorem 4.2. However, the limit self-shrinker is not L-stable by Colding-Minicozzi's theorem (c.f. Theorem 3.7) and we obtain a contradiction.

Convergence away from singularities
We recall Ilmanen's local Gauss-Bonnet formula in [41] to control the L 2 norm of the second fundamental form. Let Σ be a smooth surface with smooth boundary ∂Σ. We denote by e(Σ) the genus of Σ which is the genus of the closed surface obtained by capping off the boundary components of Σ by disks.
(c.f. Ilmanen [41]) Let R > 1 and let Σ be a surface properly immersed in B R (p). Then for any ǫ > 0 we have For simplicity, we introduce the following definition.
Note that the space C(D, N, ρ) is compact in the smooth topology by Colding-Minicozzi [18].
The following result shows that the rescaled mean curvature flow converges locally smoothly to a self-shrinker away from singularities. (1). For any T > 1, there is a subsequence, still denoted by {t i }, such that {Σ t i +t , −T < t < T } converges in smooth topology, possibly with multiplicities, to Σ ∞ away from S; (2). For any R > 0, S ∩(B R (0)×(−T, T )) consists of finite many σ-Lipschitz curves with Lipschitz constant σ depending on T and R; (3). The convergence in part (1) is also in (extrinsic) Hausdorff distance; (4). The limit self-shrinker Σ ∞ is independent of the choice of T . In other words, for different T we can choose two different subsequences of {t i } such that the corresponding flows in part (1) have the same limit self-shrinker Σ ∞ .
Proof. We divide the proof into the following steps.
Step 1. The area ratio along the flow (3.1) is uniformly bounded from above. In fact, we rescale the flow (3.1) by such that {Σ s , 0 ≤ s < 1} is a mean curvature flow satisfying the equation (1.1). By Lemma 2.9 of Colding-Minicozzi [19] and Lemma 2.3 in Li-Wang [33], we have that the area ratio of (3.1) is uniformly bounded from above.
Step 2. For any large R , the energy of Σ t ∩ B R (0) is uniformly bounded along the flow (3.1). In fact, by Lemma 3.2 we have (3.4) where N denotes the upper bound of the area ratio. Therefore, for any t > 0 the energy of Σ t ∩ B R (0) is bounded by a constant C(N, Λ, R, e(Σ)).
Step 3. For each sequence t i → +∞, we obtain a refined sequence converging to a limit selfshrinker. For any sequence t i → +∞, we rescale the flow Σ t by such that for each i the flow {Σ i,s , 1 − e t i ≤ s < 1} is a mean curvature flow satisfying (1.1) with the following properties: (a). For any small λ > 0, the mean curvature ofΣ i,s satisfies For any large R, the energy ofΣ i,s ∩ B R (0) is uniformly bounded ; (c). The area ratio is uniformly bounded from above; (d). The area ratio is uniformly bounded from below; (e). There exists a constant D ′ > 0 such that d(Σ i,s , 0) ≤ D ′ for any i.
In fact, Property (a) and (e) follow from the assumption (3.2), and Property (b) follows from (3.4). Property (c) follows from Step 1, and Property (d) follows from Lemma 3.5 in Li-Wang [33]. To prove Property (f ), by Huisken's monotonicity formula along the rescaled mean curvature flow This implies that Then (3.6) follows from (3.5) and (3.8). Therefore, by Definition 2.7 for any T > 0, small λ > 0 and any Step away from a space-time singular set S with Here s = 1 − e −t ′ . Now we show the Lipschitz property of S. By (3.5), for any curve ξ(t ′ ) of S, we can find a curve ξ(s) ofS such that where we used the inequality Note that the Lipschitz constant in (3.13) is given by The convergence is also in extrinsic Hausdorff distance by Proposition 2.8 and the limit self-shrinker is independent of the choice of T by the argument of Claim 4.3 of [33]. The lemma is proved.

Decomposition of spaces
In this subsection, we follow the argument in [33] to decompose the space and define an almost "monotone decreasing" quantity, which will be used to select time slices such that the limit selfshrinker is L-stable. First, we decompose the space as follows.
(2). The ball B R (0) can be decomposed into three parts as follows: • the high curvature part H, which is defined by • the thick part TK, which is defined by • the thin part TN, which is defined by As in Colding-Minicozzi [18], we define the L-stability of a self-shrinker.
where L Σ is the operator on Σ defined by The subindex Σ in L Σ will be omitted when it is clear in the context. We say Σ is not L-stable in the ball Recall Colding-Minicozzi's result: [18] [19])There are no L-stable smooth complete self-shrinkers without boundary and with polynomial volume growth in R n+1 .
As a corollary of Theorem 3.7, we have the following result.
Proof. For otherwise, we can find a sequence R i → +∞ and self-shrinkers . Note that f i converges smoothly to the identity map on Ω as i → +∞ and for large i its inverse map : Ω i → Ω exists and is also smooth. Moreover, f −1 i also converges smoothly to the identity map on Ω as i → +∞. We define the function ϕ i : Thus, for large i we have The lemma is proved. x ∈ Σ, we define r Σ (x) the supreme of the radius r such that where n(x) denotes the normal vector of Σ at x. Then there exists ǫ 0 (R, D, N, ρ) > 0 such that for any Σ ∈ C(D, N, ρ) and x ∈ Σ ∩ B R (0) we have Proof. We divide the proof into several steps.
Step 1. For otherwise, we can find a sequence of Σ i ∈ C(D, N, ρ) and points Let Step 2. Since x i → x ∞ , we can choose r ′ sufficiently small such that for all large i the projection of On the other hand, |s i | → 0 and for large i we have Step 3. We show that i and Σ (1) i are defined by By the smooth convergence of Σ i to Σ ∞ and the choice of On the other hand, (3.20) and (3.24) imply that where we used the fact that Σ The lemma is proved.
A direct corollary of Lemma 3.9 is the following result.
Proof. We choose ǫ 0 the same constant in Lemma 3.9. Thus, (3.29) follows from Lemma 3.9 and the definition of TN.
Using Lemma 3.10, we show that the quantity |TN| along the flow will tend to zero. Proof. By Lemma 3.4, for any t i → ∞ there exists a subsequence, still denoted by {t i }, such that it converges locally smoothly to a limit self-shrinker Σ ∞ ∈ C(D, N, ρ) away from the singular set S 0 ⊂ R 3 . For any ǫ > 0, by Definition 3.5 we have where ǫ ∈ (0, ǫ 0 ) and ǫ 0 is the constant in Lemma 3.10. The lemma is proved.
As in Lemma 4.7 of [33], we have Lemma 3.12. Fix D, R > 0 and τ ∈ (0, 1). Let {t i } be any sequence as in Lemma 3.4. If the multiplicity of the convergence in Lemma 3.4 is more than one, then for any ǫ > 0, there exists i 0 > 0 such that for any Proof. Since Σ t is embedded and {Σ t i +t , −τ ≤ t ≤ τ } converges locally smoothly to the limit self-shrinker Σ ∞ , all components of Using Lemma 3.11 and Lemma 3.12, we have the following result as in Lemma 4.8 of [33].
Proof. By Lemma 3.12, we can find . If no such time exists, then we set t 1 = s 1 . Otherwise, we choose such a time and denote it by s We denote by t 1 = s (k 1 ) 1 . After we find t 1 , set s (0) 2 = t 1 + l + 1 and continue the previous process to find time in [s Then we define t 2 = s (k) 2 . Inductively, after we find t l , we set s . This process is well defined. Repeating this process and we can find a sequence of times {t i } such that for any t i the inequality (3.30) holds. The lemma is proved.

Construction of auxiliary functions
In this subsection, we construct functions which will be used to show the L-stability of the limit selfshrinker. We fix R, T > 1 in this section. For any sequence t i → +∞, by Lemma 3.4 a subsequence of {Σ i,t , −T < t < T } converges in smooth topology to a self-shrinker Σ ∞ away from a locally finite, σ-Lipschitz singular set S ⊂ R 3 × (−T, T ). We denote by S t = {x ∈ R 3 | (x, t) ∈ S} the singular set in R 3 at time t. By Lemma 2.9, we assume that the multiplicity of the convergence is a constant N 0 ≥ 2. As in [33], we construct some functions as follows: (1). Let ǫ > 0 and large R > 0. We define and for any time interval I ⊂ (−T, T ) we define is a union of graphs over the set Ω ǫ,R (t) for large t i and any t ∈ (−T, T ).
(2). Let u + i (x, t) and u − i (x, t) be the graph functions representing the top and bottom sheets ( which we denote by Σ + i,t and Σ − i,t respectively) over Σ ∞ ∩ B R (0). The readers are referred to [33] for the details on the construction of u + i (x, t) and u − i (x, t). By the convergence property of the flow {(Σ i,t , x i (t)), −T < t < T }, for any ǫ > 0 and large R there exists i 0 > 0 such that for any i ≥ i 0 and any t ∈ (−T, T ) the functions u + i (x, t) and u − i (x, t) are well-defined on Ω ǫ,R (t). By the calculation in Appendix C, the function which we call the height difference function of Σ i,t over Σ ∞ , satisfies the equation for any (x, t) ∈ Ω ǫ,R (I) × I. Here ∆ 0 denotes the Laplacian operator on Σ ∞ . The coefficients a pq i , b p i and c i are small on Ω ǫ,R (I) × I as t i large and tend to zero as t i → +∞.
Note that the construction of the function w i is slightly different from that of [33]. In (3.35) we choose a sequence of points {x i } ⊂ Σ ∞ \S 1 to normalize the function u i , while in [33] we choose a fixed point x 0 . The reason why we choose such a normalization is that we need the inequality (3.73) in Lemma 3.19 below.
As in [33], we have the following result which implies that for large t i the integral of u i is comparable to the set |TN|. Note that (3.37) doesn't hold as ǫ → 0 since the function u i is not defined near the singularities. Lemma 3.14.
(c.f. [33]) Fix ǫ, R and T as above. For any sequence {t i } chosen in Lemma 3.4, there exists t T > 0 such that for any t ∈ (−T, T ) and t i > t T we have where dµ ∞ denotes the volume form of Σ ∞ .
Since w i satisfies the parabolic equation (3.36), we have the following parabolic Harnack inequality by using Theorem A.5 in Appendix A. (3.38) Proof. We divides the proof into several steps: Step 1. Since S t ∩ B R (0) consists of finitely many points, we can choose sufficiently small Let N be a positive integer satisfying Then τ 0 = a and τ N = b. By (3.40) we have s ≥ τ 1 . Note that (3.39) and (3.41) imply that for any k = 1, 2, · · · , N − 1 we have Step 2. Let Then we have Ω ′ ⊂ Ω ′′ ⊂ Ω. Clearly, Ω ′′ has a positive distance δ = δ(ǫ) away from the boundary of Ω. SineΩ ′ is compact, we can cover Ω ′ by finite many balls contained in Ω ′′ with radius r = ǫ 100 and the number of these balls is bounded by a constant depending only on ǫ, R and Σ ∞ . Since w i satisfies the parabolic equation (3.36), applying Theorem A.5 in Appendix A for the function w i , the domains Ω ′ , Ω ′′ , Ω and the interval [τ k−1 , τ k+1 ], we have where consists of finitely many Lipschitz curves, there exists a sequence of points {z k } such that Step 3. For s, t ∈ (a, b) with s < t, there exist integers k s and k t such that s ∈ [τ ks , τ ks+1 ) and t ∈ (τ kt , τ kt+1 ]. Note that (3.40) implies On the other hand, (3.41) implies that Step 4. Set Then by (3.39) we have
For any fixed ǫ, R and T , the following result shows that we can find a sequence {t i } such that the functions w i are uniformly bounded on a compact set away from singularities. Note that we have no estimates of w i near the singularities.
Proof. By the assumption, we can assume that K ⊂ Ω ǫ ′ ,R (I) and We divide the rest of the proof into several steps.
Step 1. w i is bounded on K × I for the time interval I = [−1, 1 2 ] and any K above. Applying Lemma 3.15 for a = −2 and b = 2 we have Step 2. w i is bounded from above on K × I for any . By Lemma 3.14 and Lemma 3.13 for large Moreover, by (3.61) we have This implies that for any t ∈ (1, On the other hand, Lemma 3.15 implies that for any where we used the fact that τ ∈ (0, 1 2 ) and Integrating the right-hand side of (3.66) and using (3.65), we have Step 3. w i (x, t) is bounded from below on K × I for any I = [a, b] ⊂ [2, T − 2] and K above. By Lemma 3.15, for any (x, t) ∈ K × I we have (3.67) In particular, for t = a the constant in (3.67) depends only on ǫ ′ , R, Σ ∞ and S [0,a+1] . Thus, the lemma is proved.
Proof. Since w i is positive by definition and w i is uniformly bounded from above by Lemma 3.16, by the interior estimates of the parabolic equation we have the space-time C 2,α estimates of w i (c.f. Theorem 4.9 of [50]), and the estimates are independent of i. Therefore, as i → +∞, the function w i converges to a limit function w in C 2 topology on K × [a, b] with w(x 0 , 1) = 1 and w is positive by the strong maximal principle. The lemma is proved.

The auxiliary functions near the singular set
In this subsection, we show that there exists a refined sequence such that the limit auxiliary function has uniform estimates across the singular set. Recall that by Lemma 3.17 the function w is uniformly bounded on any compact set away from the singular set and w has no estimates near the singularities.
In this section, we will use Lemma 3.13 repeatedly for a sequence {ǫ i } decreasing to zero, and after taking a diagonal subsequence we can construct a auxiliary function which has uniform estimates across the singular set. (1). For any k ∈ N sup (2). For any T > 0, {Σ t i,k +s , −T < s < T } converges locally smoothly to a self-shrinker Σ i,∞ ∈ C(D, N, ρ) away from the space-time singular set S i as k → +∞.
(3). For large k the surface Σ t i,k +s can be written as a union of graphs over Σ i,∞ away from the singular set S i,s . We denote byũ + i,k (x, s),ũ − i,k (x, s) the graph functions of the top and bottom sheets of Σ t i,k +s overΩ i,ǫ,R (s), wherẽ (3.70) x, s) be the height difference function of Σ t i,k +s overΩ i,ǫ,R (s). These functions are constructed as in Section 3.3. By Lemma 3.14, we can choose k i large such that for any k ≥ k i , (3.71) (4). By the smooth compactness of C(D, N, ρ) in [18], we assume that Σ i,∞ in item (2) converges smoothly to Σ ∞ ∈ C(D, N, ρ).

(5). For any
s i +s , −T < s < T } converge locally smoothly to the same self-shrinker Σ ∞ as in item (4) away from the spacetime singular set S ∞ . Moreover, the singular set S i in item (2) converges to S ∞ in Hausdorff distance.
To prove item (5), we first note that the convergence in item (2) is also in Hausdorff distance by Lemma 3.4, for any i there exists k i > 0 such that for any k ≥ k i and s ∈ (−2, 2) we have where d H denotes the Hausdorff distance. By item (4), we assume that Σ i,∞ converges smoothly to Σ ∞ ∈ C(D, N, ρ). By Lemma 3.4 for any sequence of times {s i } ∞ i=1 with s i > k i the surfaces {Σ t i,s i +s , −T < s < T } converge locally smoothly to a self-shrinker, which is denoted byΣ ∞ , away from a singular set S s ⊂ Σ ∞ as i → +∞. Moreover, as i → +∞, where we used (3.72). Thus, S i,s ∩B R (0) converges to S ∞ ∩B R (0) as i → +∞. The lemma is proved. (1). x i,k → x 0 as i → +∞ and k → +∞.
(2). For each i, there exists k i > 0 such that for any k ≥ k i , Here u i,k denotes the height difference function of Σ t i,k +s over Σ ∞ .
Proof. Choose x 0 ∈ (Σ ∞ \S 1 ) ∩ B R (0) and we denote by l x 0 the normal line of Σ ∞ passing through the point x 0 . Then the set Σ i,k ∩ l x 0 is nonempty for large i and k. Since Σ i,k can be viewed as a union of multiple graphs over Σ i,∞ away from singularities, we assume that l x 0 intersects with the bottom sheet of Σ i,k at the point y i,k , and the projection of y i,k on Σ i,∞ is x i,k ∈ Σ i,∞ . We denote by l x i,k the normal line of Σ i,∞ passing through the point x i,k . Since x 0 ∈ S 1 , we have x i,k ∈ S i,1 for large i and k. By the construction of x i,k , it is clear that x i,k converges to x 0 as i → +∞ and k → +∞. Fix θ 0 ∈ (0, π 2 ). Since Σ i,∞ converges smoothly to Σ ∞ , the angle between the two lines l x 0 and l x i,k will lie in [0, θ 0 ) for large i and there is a uniform r 0 > 0 independent of i such that We assume that Σ ′ is Σ i,∞ , Σ ′ u 1 is the top sheet of Σ i,k , Σ ′ u 2 is the bottom sheet of Σ i,k and P is the point x i,k as above. Then we apply Lemma 3.24 below for such Σ ′ , Σ ′ u 1 , Σ ′ u 2 and the point P and we can get that the functionsũ i,k and u i,k satisfy (3.73) for large k. The lemma is proved.
For s, t ∈ (a, b) with s < t, there exist integers k s and k t such that s ∈ [τ ks , τ ks+1 ) and t ∈ (τ kt , τ kt+1 ]. Then we have k t − k s ≥ 4. where we used (3.85). Similar to the proof of (3.87), we havê Combining this with (3.87)-(3.89), we havê The next result shows that the normalized height difference functionw i,k has uniformly L 1 estimate away from the singular set near t = 0, and the estimate doesn't depend on i. The proof of this result relies on the growth estimates ofw i,∞ near the singular set, which is given in Theorem 4.2 in the next section. Lemma 3.21. Fix τ ∈ (0, 1 2 ). Under the same assumptions as in Lemma 3.18, for each i we can choose k i sufficiently large such that for any k ≥ k i the normalized height difference functioñ where the points {x i,k } are chosen as in Lemma 3.19, satisfies the inequality Here W 0 is a constant independent of i.
Note that by Lemma 3.17 for each i the functionw i,k converges in C 2 to the limit functionw i,∞ away from S i and x i,k → x i,0 as k → +∞. Applying Theorem 4.2 to the functionŵ i =w i,∞ e − |x| 2 8 , we obtain that there exist uniform constants C = C(ρ 0 , Ξ 0 , R) and r 1 (ρ 0 , Ξ 0 , R) > 0 such that where K i is a compact set defined by where d g i denotes the intrinsic distance function of (Σ i,∞ , g i ). For any t ∈ (−T, T ), we define Since (Σ i,∞ , g i ) converges smoothly to (Σ ∞ , g ∞ ) and S i converges to S ∞ by Lemma 3.18, for any t ∈ (−T, T ) K i,r (t) converges smoothly to a limit set, which we denote by K ∞,r (t) ⊂ Σ ∞ . By part (5) of Lemma 3.18, K ∞,r (t) ∩ S t = ∅. Note that K ∞,r (t) is defined with respect to the metric g ∞ while Ω r,R (t) is with respect to the Euclidean metric in R 3 . Let where d(x, p) denotes the Euclidean distance in R 3 . Thus, we have Since K i,r 1 (t) andΩ i,r ′ 1 ,R+1 (t) converge to K ∞,r 1 (t) and Ω r ′ 1 ,R+1 (t) respectively for each t, for large i we have where we used the fact thatw i,∞ (x i,0 , 1) = 1. Integrating both sides of (3.98) on K i , we have Thus, the L 1 norm ofw i,∞ is uniformly bounded. Sincew i,k converges tow i,∞ on any compact set away from singularities as k → +∞, we can choose k i large such that for any k ≥ k i , where we used the inequality (3.100). Thus, the inequality (3.92) is proved.
Combining Lemma 3.18, Lemma 3.19 with Lemma 3.21, we have the following result.
Lemma 3.22. Let R > 0 and τ ∈ (0, 1 2 ). There is a sequence of times t i → ∞, a self-shrinker Σ ∞ ∈ C(D, N, ρ), a locally finite singular set S, and a constant W satisfying the following properties.
(1). For any T > 1, there exists a subsequence {t i k } ∞ k=1 of {t i } such that {Σ t i k +s , −T < s < T } converges locally smoothly to Σ ∞ ∈ C(D, N, ρ) away from S; (2). Let x 0 ∈ Σ ∞ \S 1 . We define the functions u i as in (3.33) and w i by For any ǫ > 0 and large t i , we have the inequality

102)
where W is a constant independent of i and T .
Proof. Fix a sequence of ǫ i → 0. We choose t i = t i,k i with k i large such that Lemma 3.19 and Lemma 3.21 hold. Note that u i (x, s) = u i,k i (x, s) is the height difference function of Σ t i,k i +s over Σ ∞ . Then for any T > 1 the sequence {Σ t i +s , −T < s < T } converges locally smoothly to Σ ∞ ∈ C(D, N, ρ) away from S. Note that the limit self-shrinker Σ ∞ is independent of the choice of T by Lemma 3.4. For any ǫ > 0, we have ǫ i ∈ (0, ǫ) for large i. Moreover, for large t i we have where we used Lemma 3.14 in the first inequality and (3.69) in the second inequality. Note that (3.71) implies that  taking T → +∞ we obtain a function, still denoted by w, on (Σ ∞ × (0, ∞))\S with the estimates (3.107)-(3.108). The proposition is proved.
The following result was used in the proof of Lemma 3.19.

This implies that
if we choose θ sufficiently small. Thus, the lemma is proved.

The L-stability of the limit self-shrinker
In this subsection, we show that the limit self-shrinker is L-stable. The rough idea is similar to that of [33], but the details are much more complicated. Compared with [33], the singularities here no longer move along straight lines, we cannot choose time large enough such that a given compact set doesn't contain the singularities(c.f. Lemma 4.13 of [33]). Therefore, we have to choose a cutoff function near the singularities and analyze the asymptotical behavior of the positive solution near the singular set. The analysis of the asymptotical behavior is very difficult and we delay the arguments in the next section.
The main result in this subsection is the following lemma.
We assume that φ(x, t) is a function satisfying the properties that for any t ∈ I we have φ(·, t) ∈ W 1,2 0 (Σ ∞,R ), Supp(ϕ(·, t)) ∩ S t = ∅. Then for any t ∈ I, we have This implies that for any t ∈ I, To get the inequality (3.122), the main difficulty is to estimate the last term of (3.124). Using a cutoff function inspired by [33], we will see that the last term of (3.124) depends on the asymptotical behavior of w near the singular set.
for a constant W . Therefore, by taking b i → +∞ and a = 2 in (3.145) we get (3.122). The lemma is proved.

Proof of Theorem 3.1
In this subsection, we show Theorem 3.1. First, using Lemma 3.9 and the compactness result of Colding-Minicozzi [18] we have the following result.
Proof. We show that there exists a constant C R = C(R, D, N, ρ) > 0 such that for all Σ ∈ C(D, N, ρ) we have sup Σ∩B R+1 (0) |A| ≤ C R . For otherwise, we can find a sequence Σ i ∈ C(D, N, ρ) such that sup On the other hand, by the compactness theorem of Colding-Minicozzi [18], Σ i converges smoothly to Σ ∞ ∈ C(D, N, ρ), which has bounded |A| on any compact set. This contradicts (3.147).
Using the uniform upper bound of the area ratio and Lemma 3.4, we have the following result.

Lemma 3.30. Under the same assumption as in Lemma 3.4, if {Σ t i } converges locally smoothly to
Σ ∞ with multiplicity m ∈ N away from a locally finite singular set S 0 , then for any (3.148) Proof. Since S 0 is locally finite, without loss of generality we assume that B r (x ∞ ) ∩ Σ ∞ consists of only one singular point y ∞ . For any ǫ > 0 by the smooth convergence of Since the area ratio is uniformly bounded from above along the rescaled mean curvature flow, we have as ǫ → 0. Taking ǫ → 0 in both sides of (3.149), we have (3.148). The lemma is proved.
Combining Lemma 3.29, Lemma 3.30 with Lemma 2.9, we show that the area ratio is always close to an integer after a fixed time.
Lemma 3.31. Fix large R and small δ 0 ∈ (0, 1 2 ). Under the same assumption as in Lemma 3.4, there exists t 0 > 0 such that for any t > t 0 we have Proof. We divide the proof into several steps.
Step 1. We show that there exists t 0 > 0 such that for any t > t 0 (3.150) holds for some integer m(x, t), which may depend on x and t. For otherwise, there exist a sequence t i → +∞ and By Proposition 2.8, by taking a subsequence if necessary we assume that Σ t i converges locally smoothly to a self-shrinker Σ ∞ ∈ C(D, N, ρ) with multiplicity m 0 ∈ N and . By the convergence of {Σ t i } and Lemma 3.30, we have Combining (3.152) with (3.153), for large t i we have which contradicts (3.151).
Step 2. We show that m(x, t) is independent of x and we can write m(t) for short. For otherwise, we can find a sequence t i → +∞ and x i , y i ∈ Σ t i with m(x i , t i ) = m(y i , t). Since m(x, t) ∈ [1, N 0 ], by taking a subsequence if necessary we assume that m(x i , t i ) = m 1 for all i. Thus, for any i we have m(y i , t i ) = m 1 .
(3.155) By Proposition 2.8, by taking a subsequence if necessary we assume that Σ t i converges locally smoothly to a self-shrinker Σ ∞ ∈ C(D, N, ρ) with multiplicity m 0 ∈ N, and Step 3. We show that m(t) is independent of t. It suffices to show that for any s ∈ (− 1 2 , 1 2 ), we have m(t) = m(t + s). For otherwise, we can find a sequence t i → +∞ and s i ∈ (− 1 2 , 1 2 ) such that for all i, m(t i ) = m(t i + s i ). Then there is a self-shrinker Σ ∞ ∈ C(D, N, ρ) such that for any T > 1 we can find a subsequence, still denoted by {t i }, such that {Σ t i +t , −T < t < T } converges in smooth topology, possibly with multiplicities at most N 0 , to Σ ∞ away from a singular set S. If the multiplicity of the convergence is greater than one, Lemma 3.25 shows that the limit selfshrinker Σ ∞ is L-stable in the ball B R (0). This contradicts Lemma 3.8. Therefore, the multiplicity is one and the convergence is smooth. We next show that for any sequence of s i → +∞ there exists a subsequence such that the multiplicity of the convergence is also one. For otherwise, there exists a sequence s i → +∞ such that Σ s i converges locally smoothly to a self-shrinker Σ ′ ∞ ∈ C(D, N, ρ) with multiplicity m ′ > 1. By Lemma 3.31, there exists t 0 > 0 such that for any t > t 0 we have where m is a positive integer independent of x and t. By taking t = t i → +∞ in (3.159), we have m = 1. On the other hand, taking t = s i → +∞ in (3.159), we have m = m ′ > 1, which is a contradiction. Thus, the theorem is proved.

Estimates near the singular set
In this section, we will study the asymptotical behavior of the function w near the singular set. These estimates are used in the proof of Lemma 3.21 and Lemma 3.28. In [7], Kan-Takahashi studied time-dependent singularities in semilinear parabolic equations along one singular curve. Here we develop Kan-Takahashi's techniques to estimate the solution when the singular sets consists of multiple singular curves. First, we introduce the following notations. Throughout this section, we denote by B r (p) the (intrinsic) geodesic ball centered at p in (M, g) and d g (x, y) the distance from x to y with respect to the metric g.  (1). for any p ∈ A, the harmonic radius at p satisfies r h (p) ≥ ρ; (2). For any p ∈ A, the ball B ρ (p) has harmonic coordinates {x 1 , x 2 , · · · , x m } such that the metric tensor g ij in these coordinates satisfies for any multi-index α with 1 ≤ |α| ≤ k.
The following theorem is the main result in this section, which gives the asymptotical behavior of a positive solution of a parabolic equation near a time-dependent singular set.

an integer chosen as in Corollary 4.4. Then we have
(1) u ∈ L 1 loc (Σ × (T 1 , T 2 )). More precisely, for any (t 1 , t 2 ) ⊂ (T 1 , T 2 ), there exists a constant (2) For any (t 1 , t 2 ) ⊂ (T 1 , T 2 ), we have We sketch the proof of Theorem 4.2. First, we show an asymptotical formula for the heat kernel on a Riemannian manifold in Theorem 4.3. Using this formula, we construct a special function U k (x, t) for each singular curve ξ k and a measure ν, and show that U k (x, t) behaves like log 1 r k (x,t) when x is near ξ k and ν is the Lebesgue measure in Lemma 4.5. Moreover, U k (x, t) satisfies the growth estimates (4.4)-(4.5) by Lemma 4.6, and we use U k (x, t) to construct a function v k in Lemma 4.5, which satisfies the backward heat equation. The function v k is important to construct some cutoff functions(c.f. Definition 4.12). When the singular curves are disjoint, using these cutoff functions we can show (4.3) directly in Lemma 4.9. When the singular curves are not disjoint, we show the finiteness of a functional I and use the functional I to show the L 1 norm of u (4.3) in Lemma 4.13. By using the functional I, we get a positive linear functional µ k for each singular curve ξ k in Lemma 4.15, and by Lemma 4.16 µ k is uniformly bounded even if the singular curves are not disjoint. Finally, we use µ k to construct U k and show that u is controlled by U k . By the properties of U k , we have that u satisfies the growth estimates (4.4)-(4.5).

Properties of the heat kernel
In this subsection, we will give the expansion of the heat kernel on Riemannian manifolds. Let (M, g) be a complete Riemannian manifold (without boundary) of dimension m. Suppose that p(x, y, t) is the heat kernel. Then p(x, y, t) has the following asymptotical formula (c.f. Theorem 11.1 of [6]) as t → 0 and d g (x, y) → 0. The next result gives more estimates on the asymptotical formula.
Proof. We follow the argument in Theorem 11.1 of [6] to prove (4.7)-(4.8). Define the function Direct calculation shows that For fixed x and y ∈ B ρ 0 2 (x), there exists a sequence of function {u i (x, y)} satisfying This implies that As in the proof of Theorem 11.1 of [6], we have that where C is a constant such that u 0 (x, x) = 1 and J(y) is the area element of the sphere of radius d g (x, y) at the point y. There exists integer k 0 depending only on k such that under the assumption Let ρ = ρ 0 4 . Now we choose a cutoff function η(r) with 0 ≤ η ≤ 1 such that η(r) = 1 when r ≤ ρ and η(r) = 0 when r ≥ 2ρ. Define χ(x, y) = η(d g (x, y)) and F (x, y, t) = χ(x, y)G(x, y, t). If d g (x, y) ≤ ρ and t ≤ 1, the identity (4.9) gives that where we used (4.10) in the last inequality. Similarly, for ρ ≤ d g (x, y) ≤ 2ρ we can also check that .
Combining the above estimates, we have where we used the fact that M p(x, y, t)dvol y ≤ 1. Thus, (4.11) gives (4.7). Similarly, we can show that p(z, y, s) dz Thus, (4.12) implies (4.8). The theorem is proved.
As a corollary, we have the following result in dimension two.
Using Corollary 4.4, we have the following result.
For t ∈ (T ,T ) with |Dν| < +∞ we write where G(s) satisfies lim s→t − G(s) t−s = 0 for a.e. t ∈ (T ,T ). Let λ ∈ (0, t − t). Note that U 0 can be written as By Theorem 4.3 I 1 satisfies I 1 ≤ 1 4πλ ν((T , t − λ)) < +∞. Thus, we have where we assumed that R is small such that | log r| ≥ 1 for any r ∈ (0, R). Moreover, we have where C 1 is a universal constant. Next, we estimate I 2 . Using Corollary 4.4, (4.30) and integration by parts we have t t−λ p 0 (x, ξ(s), t − s) ds where we can choose c = 1 and C 2 is a constant depending on σ and λ. Therefore, we have 1 , t 2 )). (4.37) Finally, we estimate I 3 . Using the inequality (4.30) for c = 1 and integration by parts, we have where C 3 depends on σ and λ. Thus, we have

Estimates of the solution with disjoint singularities
In this subsection, we follow Section 4.1 of [7] to construct some cutoff functions and show the integrability of the solution across the singular set when the singular curves are disjoint. First, we construct some cutoff functions. (1). Let t 3 < t 1 < t 2 < t 4 and 0 < δ < r 1 . Define ζ = ζ(t; t 1 , t 2 , t 3 , t 4 , δ, r 1 ) ∈ C ∞ (R) such this that (4.40) (2) Let η be a smooth function on R satisfying and define H(z) = z 0 η(τ ) dτ . Then H(z) satisfies the inequality We keep the same notation H(z) as in [7]. Throughout this section, H always denotes the function as above and it should not be confused with the mean curvature.
Consider the case that there is only one singular curve. We show that the solution of (4.2) is in L 1 across the singular set. The argument is the same as that of [7] and we give all the details for the readers' convenience.

By (4.42) and (4.47) we have
Note that where we used the construction of ζ(t) in Definition 4.7. Thus, we have where c φ,K = sup K (|∂ t φ| + |∆φ| + |∇φ|). Combining the above estimates, we have Combining (4.53) (4.54) with (4.51), we have Taking r 0 sufficiently small and using the assumption that c(x, t) is locally bounded, we have Therefore, by the definition of φ we have Note that the function H ′ • w converges to 1 on Q r 1 ,t 1 ,t 2 \Γ t 1 ,t 2 as δ → 0. Thus, taking δ → 0 in (4.55), we have that u is integrable on Q r 1 ,t 1 ,t 2 . The lemma is proved.
Summarizing the above discussion, we define We call that the singular curves {ξ 1 , ξ 2 , · · · , ξ l }(t ∈ I) are around (x 0 , t 0 ) on I, if the curves satisfy the conditions in Case (A) or are constructed as in Case (B) on I.
We construct some cutoff functions when the singular curves are not disjoint.
As a byproduct of the above proof, we have the following result. I ξ (ρ; t 1 , t 2 , u, 2r 1 ) < +∞. By using Lemma 4.9, Lemma 4.14 and following the same arguments as in [7], we have the following results when the singular curves are disjoint. (c.f. [7]) Under the same assumption as in Theorem 4.2, if we assume that {ξ 1 (t), · · · , ξ l (t)} are disjoint on [T 1 , T 2 ] andρ is the constant in (6) of Definition 4.12, then we have (1). For each ξ k and (t 1 , t 2 ) ⊂ [T 1 , T 2 ], the mapping J k : defines a distribution whose support is contained in Γ (k) t 1 ,t 2 , and satisfies where C is a universal constant. Here I ξ k (ρ; t 1 , t 2 , u,ρ) is defined in (6) of Definition 4.12 and it is finite by Lemma 4.14.
Proof. Since {ξ 1 (t), · · · , ξ l (t)} are disjoint on [t 1 , t 2 ], we can consider each ξ k as in [7]. After replacing the function g in (4.21) of [7] by the function c(x, t), we know that part (1) follows directly from Lemma 4.4 of [7]. Part (2) follows from the proof of Theorem 2.1 of [7](See Page 7303 of [7]), (3) follows from Lemma 5.2 of [7] and (4) follows from the non-negativity of the right-hand side of (4.109). Since the proof is exactly the same as in [7], we omit the details here.
When the singular curves are around (x 0 , t 0 ), the measures µ k constructed in Lemma 4.15 may blow up as t → t 0 . The next result shows that µ k is actually bounded when t is close to t 0 .

Proof of main theorems
In this section, we prove Theorem 1.2 and Corollary 1.4.
Proof of Theorem 1.2. Suppose that the mean curvature flow (1.1) reaches a singularity at (x 0 , T ) with T < +∞. Then Corollary 3.6 of [25] implies that for all t < T we have (2) the mean curvature ofΣ s satisfies |H(p, s)| ≤ Λ 0 for some Λ 0 > 0; Fix τ > 0. By Theorem 3.1, for any sequence s i → +∞ there exists a subsequence, still denoted by {s i }, such that the flow {Σ s i +s , −τ < s < τ } converges smoothly to a self-shrinker with multiplicity one. In other words, taking c j = e Therefore, for fixed τ > 0 the flow {Σ j t , −e τ <t < −e −τ } converges smoothly to a smooth selfshrinker flow with multiplicity one as j → +∞. Theorem 1.2 is proved.
Proof of Corollary 1.4. We follow the argument in the proof of Theorem 1.2. Suppose that Then the rescaled mean curvature flow (5.2) satisfies |H| ≤ δ 0 . There exists a sequence of times s i → +∞ such that for any fixed τ > 0 the flow {Σ s i +s , −τ < s < τ } converges smoothly to a self-shrinker Σ ∞ ∈ C(2, N, ρ) with multiplicity one. Moreover, the mean curvature of the limit self-shrinker satisfies sup Σ∞ |H| ≤ δ 0 . On the other hand, we have with sup Σ i |H| ≤ δ i → 0. By the smooth compactness result of self-shrinkers in [18], we can assume that Σ i converges smoothly to a self-shrinker Σ ∞ ∈ C(D, N, ρ) with multiplicity one. Since the convergence is smooth, the limit self-shrinker Σ ∞ has zero mean curvature and by Corollary 2.8 of [19] it must be a plane passing through the origin. For any hypersurface Σ n ⊂ R n+1 , the F -functional of Σ is defined by Colding-Minicozzi [19] Moreover, the entropy λ of Σ in [19] is defined by where the supremum is taken over all t 0 > 0 and x 0 ∈ R n+1 . By Lemma 7.10 of [19], the entropy λ is achieved by (x 0 , t 0 ) = (0, 1) for a self-shrinker with polynomial volume growth. Thus, we have Since Σ i converges smoothly to a plane Σ ∞ with multiplicity one, by (5.7) we have Thus, for any ǫ > 0 and large i we have λ(Σ i ) < 1 + ǫ. By Theorem 1.2 of C. Bao [8] Σ i must be a plane, or we can use Guang-Zhu's rigidity result in [32]. Note that for any large R, Σ i ∩ B R (0) is graphic over Σ ∞ ∩ B R (0) for large i. Together with λ(Σ i ) < 1 + ǫ < 2, we have that Σ i is a plane by Theorem 0.1 of [32]. This contradicts our assumption that Σ i is non-flat. The lemma is proved.
Therefore, by Lemma 5.1 the limit self-shrinker Σ ∞ must be a plane passing through the origin. Consider the Heat kernel function Thus, Huisken's monotonicity formula(c.f. Theorem 3.1 in [37]) implies that which implies that (x 0 , T ) is a regular point by Theorem 3.1 of White [60]. Thus, the flow {(Σ, x(t)), 0 ≤ t < T } cannot blow up at (x 0 , T ). The corollary is proved.

Appendix A Krylov-Safonov's parabolic Harnack inequality
In this appendix, we include the parabolic Harnack inequality from Krylov-Safonov [48]. First, we introduce some notations. Let x = (x 1 , x 2 , · · · , x n ) ∈ R n . Denote Consider the parabolic operator where the coefficients are measurable and satisfy the conditions Here b(x, t) = (b 1 (x, t), · · · , b n (x, t)). Then we have Moreover, when 1 θ−1 and 1 µ vary within finite bounds, C also varies within finite bounds.
Note that in our case the equation (3.36) doesn't satisfy the assumption that c(x, t) ≥ 0 in (A.4). Therefore, we cannot use Theorem A.1 directly. The following result shows that the Harnack inequality still works when c(x, t) is bounded.
) is a nonnegative solution to the equation where the coefficients a ij (x, t) and b i (x, t) satisfy (A.2)-(A. 3), and c(x, t) satisfies Then there exists a constant C, depending only on θ, µ and n, such that Proof. Since u(x, t) is a solution of (A.6) and c(x, t) satisfies (A.7), the function v( Applying Theorem A.1 to the equation (A.9), we have where C depends only on θ, µ and n. Thus, for any x ∈ B R 2 (0) we have where C ′ depends only on θ, µ and n. Here we used R ≤ 2 by the assumption. The theorem is proved.
We generalize Theorem A.2 to a general bounded domain in R n .
Let Ω be a bounded domain in R n . Suppose that u(x, t) ∈ W 1,2 n+1 (Ω × (0, T )) is a nonnegative solution to the equation where the coefficients a ij (x, t) and b i (x, t) satisfy (A.2)-(A. 3), and c(x, t) satisfies (A.7) for a constant µ > 0. For any s, t satisfying 0 < s < t < T and any x, y ∈ Ω with the following properties (1). x and y can be connected by a line segment γ with the length |x − y| ≤ l; (2). Each point in γ has a positive distance at least δ > 0 from the boundary of Ω; (3) s and t satisfy T 1 ≤ t − s ≤ T 2 for some T 1 , T 2 > 0; we have u(y, s) ≤ C u(x, t), where C depends only on n, µ, min{s, δ 2 }, l, T 1 and T 2 .
Proof. Let γ be the line segment with the property (1) and (2) connecting x and y. We set Here we choose N to be the smallest integer satisfying We can check that R ≤ δ 2 . For any s, t ∈ (0, T ), we choose {t i } N i=0 such that t 0 = s, t N = t and for all integers 1 ≤ i ≤ N . Note that (A.13)-(A.15) imply that for any 0 ≤ i ≤ N − 1, Therefore, for any 0 ≤ i ≤ N − 1 we have (t i+1 − θR 2 , t i+1 ) ⊂ (0, T ) and p i+1 ∈ B R 2 (p i ). Applying where C depends only on c, n, µ and 1 θ−1 = R 2 N t−s . Here we used the fact that t i = (t i+1 − θR 2 ) + R 2 . Therefore, u(y, s) = u(p 0 , t 0 ) ≤ C N u(p N , t N ) = C ′ u(x, t) (A. 17) where the constant C ′ in (A.17) depends only on c, n, µ, min{s, δ 2 }, l, T 1 and T 2 . The theorem is proved.
A direct corollary of Theorem A.3 is the following result.
Then for any s, t satisfying 0 < s < t < T and any x, y ∈ Ω ′ , we have u(y, s) ≤ C u(x, t), (A. 19) where C depends only on n, µ, min{s, δ 2 }, t − s, r and k.
Proof. By the assumption, we can find finite many points A = {q 1 , q 2 , · · · , q k } such that For any x, y ∈ Ω ′ , there exists two points in A, which we denote by q 1 and q 2 , such that x ∈ B r (q 1 ) and y ∈ B r (q 2 ). Then x and y can be connected by a polygonal chain γ, which consists of two line segments xq 1 , yq 2 and a polygonal chain with vertices in A connecting q 1 and q 2 . Clearly, the number of the vertices of γ is bounded by k + 2 and the total length of γ is bounded by (k + 2)r. Moreover, by the assumption we have γ ⊂ Ω ′′ and each point in γ has a positive distance at least δ > 0 from the boundary of Ω.
Assume that the polygonal chain γ has consecutive vertices {p 0 , p 1 , · · · , p N } with p 0 = y, p N = x and 1 ≤ N ≤ k + 2. We apply Theorem A.3 for each line segment p i p i+1 and the interval [t i , t i+1 ], where {t i } is chosen as in (A.15). Note that Thus, for any 0 ≤ i ≤ N − 1 we have u(p i , t i ) ≤ Cu(p i+1 , t i+1 ), (A. 21) where C depends only on c, n, µ, min{s, δ 2 }, r, k and t − s, and (A.21) implies (A.23). This finishes the proof of Theorem A.4.
Theorem A.4 can be generalized to Riemannian manifolds by using the partition of unity. Here we omit the proof since the argument is standard. Note that the constant in (A.23) depends on the geometry of (M, g).
Proof. By the assumption on Ω ′ , Ω ′′ and Ω, x and y can be connected by a path γ in Ω ′′ with bounded length and every point in γ has a distance at least δ from the boundary of Ω. Thus, the theorem follows directly from Theorem B.1 by choosing R = δ and α = 2.
In the proof of Lemma 3.21, we need to use Theorem B.2 to a class of surfaces with bounded geometry. In order to show that the constants in the Harnack inequality is uniformly bounded, we have the following result.
Theorem B.3. Fix R > 0. We assume that (1). Σ 2 i ⊂ R 3 is a sequence of complete surfaces which converges smoothly to a complete surface Σ in R 3 ; (2). The Ricci curvature of Σ∩B R (0) is bounded by a constant −K with K ≥ 0. Here B R (0) ⊂ R 3 denotes the extrinsic ball centered at 0 with radius R; ( 3).
Proof. It suffices to show that Ω ′ i , Ω ′′ i and Ω i satisfy the (δ ′ , k ′ , r ′ ) property of Theorem B.2 with uniform constants δ ′ , k ′ and r ′ . By the smooth convergence of Ω i to Ω, we define the map ϕ i : Ω → Ω i by ϕ i (x) = x + u i (x)n(x), ∀ x ∈ Ω, (B.11) where u i (x) is the graph function of Ω i over Ω and n(x) denotes the normal vector of Σ at x. Note that ϕ i (Ω) = Ω i and ϕ i converges in C 2 to the identity map on Ω as i → +∞. By the assumption (5), there exists k points {p α } k α=1 ⊂ Ω ′ and ǫ > 0 such that where Ω ′′ 4ǫ = {x ∈ Ω ′′ | d Σ (x, ∂Ω ′′ ) ≥ 4ǫ}. Therefore, we have Since the C l norms of u i in (B.11) are small, for large i we have ϕ i (B r (p α )) ⊂ B i,r+ǫ (ϕ i (p α )) ⊂ Ω ′′ i,2ǫ , (B.14) where Ω ′′ i,2ǫ = {x ∈ Ω ′′ i | d Σ i (x, ∂Ω ′′ i ) ≥ 2ǫ} and B i,r (p) denotes the geodesic ball of Σ i centered at p with radius r. Combining (B.13) with (B.14), we have Therefore, Ω ′ i can be covered by k geodesic balls with radius r + ǫ, and all balls are contained in Ω ′′ i . It is clear that Ω ′′ i has a positive geodesic distance δ 2 > 0 from the boundary of Ω i for large i. Thus, Ω ′ i , Ω ′′ i and Ω i satisfy the ( δ 2 , k, r + ǫ) property and the theorem follows directly from Theorem B.2.

Appendix C The linearized equation of rescaled mean curvature flow
In this appendix, we follow the calculation in Appendix A of Colding-Minicozzi [21] to show (3.34). See also Appendix A of Colding-Minicozzi [22]. Let Σ be a hypersurface in R n+1 and Σ u the graph of a function u over Σ. Then Σ u is give by where n(x) denotes the normal vector of Σ at x. We assume that |u| is small. Let e n+1 be the gradient of the signed distance function to Σ and e n+1 equals n on Σ. We define ν u (p) = det g u ij (p) det g ij (p) , w i (p) = e n+1 , n u , η u (p) = p + u(p)n(p), n u (C. 2) where g ij denotes the metric on Σ at p, g u ij is the induced metric on Σ u and n u is the normal to Σ u .